Let $\mathrm{a}: \sim(\mathrm{p} \wedge \sim \mathrm{r}) \vee(\sim \mathrm{q} \vee \mathrm{s})$ and…

Let $\mathrm{a}: \sim(\mathrm{p} \wedge \sim \mathrm{r}) \vee(\sim \mathrm{q} \vee \mathrm{s})$ and $\mathrm{b}:(\mathrm{p} \vee \mathrm{s}) \leftrightarrow(\mathrm{q} \wedge \mathrm{r})$. If the truth values of a $p$ and $q$ are true and that of $r$ and $s$ are false, then the truth values of a and b are respectively
  1. T,F
  2. T,T
  3. F,F
  4. F,T

Solution

We have $\mathrm{p}, \mathrm{q} \equiv \mathrm{T}$ and $\mathrm{r}, \mathrm{s} \equiv \mathrm{F}$ $\begin{aligned} & \mathrm{a}: \sim(\mathrm{p} \wedge \sim \mathrm{r}) \vee(\sim \mathrm{q} \vee \mathrm{s}) \equiv \sim(\mathrm{T} \wedge \sim \mathrm{F}) \vee(\sim \mathrm{T} \vee \mathrm{F}) \\ & \equiv \sim(\mathrm{T} \wedge \mathrm{T}) \vee(\mathrm{F} \vee \mathrm{F}) \\ & \equiv \sim \mathrm{T} \vee \mathrm{F} \equiv \mathrm{F} \vee \mathrm{F} \equiv \mathrm{F} \\ & \mathrm{b}:(\mathrm{p} \vee \mathrm{s}) \leftrightarrow(\mathrm{q} \wedge \mathrm{r}) \equiv(\mathrm{T} \vee \mathrm{F}) \leftrightarrow(\mathrm{T} \wedge \mathrm{F}) \equiv \mathrm{T} \leftrightarrow \mathrm{F} \equiv \mathrm{F} \end{aligned}$

Asked in: MHT CET 2021 (22 Sep Shift 2)

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